DiracSolver: A tool for solving the Dirac equation

Advantageous numerical methods for solving the Dirac equations are derived. They are based on different stochastic optimization techniques, namely the Genetic algorithms, the Particle Swarm Optimization and the Simulated Annealing method, their use of which is favored from intuitive, practical, and theoretical arguments. Towards this end, we optimize appropriate parametric expressions representing the radial Dirac wave functions by employing methods that minimize multi parametric expressions in several physical applications. As a concrete application, we calculate the small (bottom) and large (top) components of the Dirac wave function for a bound muon orbiting around a very heavy (complex) nuclear system (the 208Pb nucleus), but the new approach may effectively be applied in other complex atomic, nuclear and molecular systems.

本文推导出求解狄拉克方程的有利数值方法，这些方法基于多种随机优化技术，包括遗传算法、粒子群优化以及模拟退火法，其应用之选基于直观、实践及理论论据。为此，我们通过最小化多参数表达式在多个物理应用中的方法，对表示径向狄拉克波函数的适当参数表达式进行优化。具体应用方面，我们计算了围绕非常重（复杂）的核系统（如208Pb核）轨道运动的束缚μ子的狄拉克波函数的小（底部）和大（顶部）成分，但此新方法亦能有效应用于其他复杂的原子、核和分子系统。